Math topic中文

Foot of perpendicular and distance

Foot of the perpendicular and distance from a point to a line or to a plane.

Practice questions

1The lines $l_1$ and $l_2$ have equations $\mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 10 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ -6 \end{pmatrix}, \lambda \in \mathbb{R}$ …
2A drilling company plans to install a straight pipeline $AB$ through a mountain. Points $(x, y, z)$ are defined relative…
3With reference to the origin $O$ , the points $A$ , $P$ and $Q$ have position vectors $\mathbf{a}$ , $\mathbf{p}$ and $\mathbf{q}$ …
4The planes $\Pi_1$ and $\Pi_2$ have vector equations $\mathbf{r} = \lambda_1(\mathbf{i} + \mathbf{j} - \mathbf{k}) + \mu_1(2\mathbf{i} - \mathbf{j} + \mathbf{k}) \quad \text{and} \quad \mathbf{r} = \lambda_2(\mathbf{i} + 2\mathbf{j} + \mathbf{k}) + \mu_2(3\mathbf{i} + \mathbf{j} - \mathbf{k})$ …
5A home interior designer is looking to design a pull-down wardrobe lift. To test his concept, the designer built a small…
6The diagram below shows a tetrahedron $ABCD$ . The equation of the plane $ABD$ is $4x + y + 2z = 16$ . Given that $A$ is o…
7The points $A$ and $B$ have coordinates $(0, 2, 0)$ and $(1, 0, 2)$ respectively, and the plane $\pi_1$ has equation $\mathbf{r} \cdot \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} = 4$ …
8A temporary isolation centre is built to manage the increasing number of COVID-19 cases. The roof takes the shape of a t…
9The equations of planes $p_1$ and $p_2$ are given by $p_1: \mathbf{r} \cdot \begin{pmatrix} \alpha \\ 7 \\ 2 \end{pmatrix} = \alpha + 20, \; \alpha \in \mathbb{R},$ …
10Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ …
11The equations of planes $p_1$ and $p_2$ are given by $p_1:\;\mathbf{r}\cdot \begin{pmatrix}\alpha\\7\\2\end{pmatrix} = \alpha + 20,\; \alpha \in \mathbb{R},$ …
12Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = a$ and $\overrightarrow{OB} = b$ …
13Relative to the origin $O$ , the points $A$ and $B$ have non-zero and non-parallel position vectors $\mathbf{a}$ and $\mathbf{b}$ …
14Planes $p$ and $q$ are perpendicular. Plane $p$ has equation $2x - z = 10$ . Plane $q$ contains the line $l$ with equatio…
15In the diagram, O is centre of the rectangular base ABCD of a right pyramid with vertex V. Perpendicular unit vectors i,…
16OABC is a rhombus. The points A and C are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$ …
17The two planes $p_{1}$ and $p_{2}$ , given by the equations $2x+y=1$ and $8x+ay+z=4$ respectively, meet at a line $L$ whi…
18The planes $p_1$ and $p_2$ have equations $\mathbf{r} = \begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix} + s \begin{pmatrix} 3 \\ -1 \\ 7 \end{pmatrix} + t \begin{pmatrix} 1 \\ -3 \\ 12 \end{pmatrix}$ …
19With reference to the origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectivel…
20In a computer game, a slope can be modelled as a plane $p$ containing three points, $A(1, 0, -3)$ , $B(1, 4, -15)$ and $C(2, 3, -14)$ …
21Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ …
22The plane $p$ passes through the points $A$ , $B$ and $C$ with coordinates $(0, 1, 1)$ , $(2, -1, 4)$ and $(-2, -1, 0)$ re…
23The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).$ …
24Referred to the origin $O$ , the points $P$ and $Q$ have position vectors $4\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$ and $8\mathbf{i} + 8\mathbf{j} - \mathbf{k}$ …
25The point A has coordinates (1, -2, 4) and the plane $\pi_1$ has equation $2x - y + 2z = 5.$ Find the exact shortest dis…
26With respect to the origin $O$ , the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respect…
27The plane $\pi_1$ contains the point $(2,-2,1)$ and the line $\mathbf{r} = \begin{pmatrix}0 \\ 1 \\ 8\end{pmatrix} + \lambda \begin{pmatrix}0 \\ 2 \\ 3\end{pmatrix},$ …
28A tetrahedron $OABC$ has four triangular faces with a vertex $O$ at the origin. The coordinates of $A$ , $B$ and $C$ are…
29The lines $l_1$ and $l_2$ have equations $\frac{x-1}{3} = \frac{y-2}{a}, z = 1 \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}, \lambda \in \mathbb{R}$ …
30The planes p and q have equations $r = (2 + \lambda - 2\mu)\,\mathbf{i} + (-3\lambda + \mu)\,\mathbf{j} + (3 - \lambda + 2\mu)\,\mathbf{k}$ …
31The plane $\pi_1$ has equation $\mathbf{r}\cdot\begin{pmatrix}2 \\ 5 \\ 1\end{pmatrix}=11.$ The plane $\pi_2$ passes t…
32Methane ( CH_4 ) is a chemical compound with a tetrahedral structure. The 4 hydrogen (H) atoms form a regular tetrahedro…
33An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. The…
34The line $l$ has equation $1 - x = \frac{z - \alpha}{2}, y = 1$ , where $\alpha$ is a constant. Given that the point of i…
35The line $l_1$ has the vector equation $\mathbf{r}=\begin{pmatrix}a\\1\\0\end{pmatrix}+\lambda\begin{pmatrix}0\\1\\b\end{pmatrix},\ \lambda\in\mathbb{R},$ …
36The equations of planes $p_1$ and $p_2$ are respectively given by $\mathbf{r} = s \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}, \quad s, t \in \mathbb{R} \quad \text{and} \quad \mathbf{r} \cdot \begin{pmatrix} \alpha \\ 3 \\ \beta \end{pmatrix} = 7,$ …
37The plane $\Pi_1$ has equation $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \theta(2\mathbf{j} - \mathbf{k}) + \phi(3\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}).$ …
38Relative to the origin, the fixed point $A$ has position vector $\mathbf{a}$ . The line $l$ has equation $\mathbf{r} = s\mathbf{d}$ …
39The plane $\Pi_1$ contains the points $A(1,4,2),\ B(1,0,5)$ and $C(0,8,-1)$ . Find a cartesian equation of $\Pi_1$ . Find…
40A designer wants to construct an inclined roof that is attached to a vertical rectangular wall, positioned above a horiz…
41Two lines, $l_1$ and $l_2$ , have equations $\frac{x - 1}{2} = -z - 1, y = 2$ and $x + 3 = 4 - y = z + 2$ respectively. D…
42The plane $p_1$ has equation $x - y + z = 0$ and the plane $p_2$ has equation $x + y + z = 2$ . Find the acute angle betw…
43The equations of two planes $P_1$ and $P_2$ are $x - 2y + 3z = 4$ and $3x + 2y - z = 4$ respectively. The equation of a…
44The line $\ell$ has the equation $\dfrac{x-a}{2}=y-2=\dfrac{1-2z}{b}$ , where $a$ and $b$ are real constants. The plane $\pi$ …
45The line l has equation $\frac{x-1}{-2}=y=\frac{z+7}{4}$ , and the plane p has equation $x-z=2$ . Find the acute angle bet…
46With reference to the origin O, the point A has position vector $-\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}$ . The plane $\pi$ …
47Relative to the origin $O$ , two fixed points, $A$ and $B$ , have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectiv…
48A player is controlling an avatar in a virtual reality game where points $(x,y,z)$ are defined relative to a fixed point…
49The planes $\pi_1$ and $\pi_2$ , which meet in the line $l_1$ , have equations $\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \text{ and } \pi_2 : x + ky - 2z = -15,$ …
50The planes $\Pi_1$ and $\Pi_2$ have equations $\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} + \mathbf{k}) = 4$ and $\mathbf{r} = 3\mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + \mathbf{k}) + \mu(4\mathbf{i} + \mathbf{j})$ …
51A tetrahedron has four vertices O, A, B and C where O is the origin. It is given that the coordinates of B are (1, 4, 6)…
52Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ …
53The diagram above shows a toy house with horizontal rectangular base $ABCD$ where $AB = 30\text{ cm}$ and $BC = 20\text{ cm}$ …
54Referred to the origin $O$ , the points $A$ , $B$ and $C$ have position vectors $3\mathbf{i} - \mathbf{j}$ , $5\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$ …
55A frigate is stationed at position $F(1, 2, 0)$ . Two submarines $S_1$ and $S_2$ are under the sea surface. Submarine $S_1$ …
56The equations of planes $P_1, P_2$ are $P_1:\quad \mathbf{r} = \begin{pmatrix}-3 \\ 10 \\ 3\end{pmatrix} + \lambda\begin{pmatrix}-2 \\ 12 \\ 3\end{pmatrix} + \beta\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix},\quad \lambda,\beta \in \mathbb{R}$ …
57The point A has coordinates (1, -2, 4) and the plane $\pi_{1}$ has equation $2x - y + 2z = 5$ . The plane $\pi_{2}$ has e…
58The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).\n\nThe points A and B on$ …
59The plane $p$ has equation $x - 2y + 2z = 12$ . With reference to the origin $O$ , the point $A$ has position vector $2\mathbf{i} + \mathbf{j} - 3\mathbf{k}$ …
60The diagram shows a solid with a horizontal rectangular base $OABC$ in which the lengths of $OA$ and $OC$ are $12\text{ m}$ …